Use language appropriate to their apparent level of comprehension of the problem. If they objectify the problem visually in a 2D diagram think very hard before asking them to project it into another representative space, or phase change or whatever is your "this is a better way to think about it"
Try very hard not to be patronising. Kids struggling with maths can be super sensitive. They can misinterpret a smile as a smirk. If they had the courage to seek help, don't belittle them. For example a phrase as simple as "that's easy" can be a huge burden.
Beware misunderstood questions. If you seek to help by asking them questions be mindful if they do not understand the intent of your question, you may add to their burden not reduce it.
Some set work requires a specific method be shown to be used. It's fine to show them other methods, but you may have to help them project the answer back into the one they will be assessed in.
If they only want debugging, then help them find the bug, the mistranscribed value, the divide by zero, the specific mistake. If they have no insight into the problem then more exploratory help may be useful.
Time out can help. Stress hormones don't make maths any easier.
Totally. Or co workers. My best source of advice refrains from pointing out a strongly typed functional language would solve most of my problems and helps the specific python syntactic burden I have right now. But then does go the distance if I engage.
After all, programming is just a subset of mathematics anyway..
> After all, programming is just a subset of mathematics anyway..
I am quite proud;-) to have once trolled a group composed of mathematics and computer science students with one sentence: "What is computer science? It's that part of mathematics which is of any use." (FTR: I'm a theoretical mathematician turned programmer.) The best part (I think) is that it has some truth to it!
There is a ton of useful applied maths that isn't computer science. There's useful abstract maths that isn't computer science too. Useful computer science is a tiny fraction of useful maths, and plenty of computer science isn't useful.
>a strongly typed functional language would solve most of my problems
I'm at the stage where I constantly notice this, which adds to my already non-zero job fatigue. I try to keep my sanity by threatening (myself) to start my own company, but maybe I just need to stop caring and grow up...
This is pretty common advice that I am starting to disagree with. In my experience (in person math and chemistry tutoring and answering programming questions on public discords) this technique really only works well with some one who is on the verge of understanding and has perhaps made a simple error or already has mastery over the techniques and concepts at play and are learning one entirely new thing.
In those cases this technique is great! You can guide them to that moment of epiphany, and it is very rewarding and efficient.
Much more commonly it feels like pulling teeth. I think this is because most kinds of misunderstandings and confusions are hard to identify from the inside. For example, when some one learns a new thing that is in contradiction with something else (that they perhaps misremembered, never fully understood, or misunderstood a nuance in a way that was internally consistent) the resulting inconsistency can be very difficult to identify and verbalize. If you can help them identify and resolve it on their own then great- my experience is that when that works it is exhausting and time consuming and that taking a more active role in identifying the conflict is much more productive.
In programming it commonly takes people years to learn enough background context that this method becomes effective. I think it is better to be quick with examples and explanations, there is just too much to know to make this sort of highly examined introspection efficient or effective.
Other times its simply because the student isn't intrinsically motivated to learn the material and this style of teaching involves a lot of emotional labor in order to be effective. Turns out people shut down a little when you ask them to explain their thought process and they feel they should not tell you frankly they care not one wit for Pythagoras or your damned flag pole .
1. I’ll tell you what the right formula/process/etc is — as a hint on the problem.
2. If you still don’t get it, I’ll walk through the right steps with you.
I’ve never found it particularly useful to dig into what you have wrong or try to guide you into self-correcting unless the problem is minor (eg, you clearly understand and just need a nudge), you ask, or I’m dealing with someone that has a professional interest and will invest the effort debugging.
I can’t imagine how badly I’d hate it if StackOverflow tried to guide me on some Socratic journey every time, rather than just tell me the correct syntax for the magic invocation of pandas that solves my problem.
True that focusing on what they're doing isn't the best course unless they're making repeatable errors.
Teaching the "tricks" can help people who already have a mindset for relationships, but often people who struggle with math don't have this, and it takes practicing fundamental methods so they can apply them.
I also found a primer in symbolic logic greatly improved student comfort and familiarity with most other materials. Knowing the formal language and rules of logic and relationships is key to being able to express and interpret mathematics.
They do. The realization that something can be applied in other contexts is what takes them from feeling overwhelmed that each step is something discrete and isolated to memorize to being excited about making it make sense.
This also happens after teaching them how to use their calculators. When they begin to be able to visualize what the approximate bounds are by understanding eg- an expression and what it applies to, it's the difference from someone who only reads word by word and someone who reads with respect to the content of the arc or chapter.
So much of math is contingent on parallels to other subjects, and I've always felt not enough was done to relate to students the similarities between the formal rules.
Ie- when teaching linguistic notation for grammar and syntax, you can have students reduce sentences to expressions (a la Backus–Naur form) and then "solve" for them using, eg- a semantic net. This is often much more comfortable to the numerically adverse, since the logic of language is often understood more intuitively. It can also be taught at any level using simple substitution, parts of speech, etc with the added benefit as a primer for its applications to machine learning.
Right? I understand why its advised so glibly - it sounds nice and when it works its powerful but..
I had one mentor that was ruthless in forcing me to explain myself and almost pedantic in his insistence on correct vocabulary at all times (programming), but I was also able to interrogate the hell out of him and had practical strategies for identifying and rooting out my own misunderstanding - it was incredibly rewarding but frequently exhausting, half the time we'd wrap up and I'd feel like my voice had grown horse from shouting.
It wouldn't have worked if we didn't trust each other, or if ever I had grown discouraged by my own ignorance, or if he hadn't worked on a compiler team and been able to describe in detail and on command the function of any part of a computer system.
And I suspect most success stories using that method are somewhat like this; deeply personal, probably shouldn't be your expectation or plan A.
The problem is that the student would spend so much time on figuring out their mistakes. If you pass the attention span of children, all they feel is that they made a mistake and feel defeated while getting lectured on something they don’t understand anyway. It is not that easy to find a mistake that you made, even I made mistakes that took me long to find out.
I’m actually in favour of the traditional way, show them the right way and let them try. It is fine to make mistakes, but don’t put focus on the mistakes. Instead focus on good practices. Asking them to figure out their own mistakes/making explanations should be done very sparsely.
Yes. The biggest hurdle is usually attention span of the child. If the child has the attention to go thru it slowly the article's approach could be great. But for cases where their attention is already shot, walking them thru the steps or demonstration of the steps can be an easier way thru. Yes they are less likely to have a major breakthru in understanding, but it can help them with memorizing the step sequence without exhausting the limited attention.
I agree with the author here for most of what was said. Except I would summarize it as "explore both the problem the student identifies and the solution the student uses" and while doing so they would catch their mistakes. Explore means to drill down and seriously make the student wonder why they're doing the problem that way.
And the wonderful thing about exploration is you don't even have to know the subject. You can just keep asking and engaging in a conversation. The student learns. And so do you.
This article and your statement is the best thing I have read all year.
This sets up a self correcting feedback loop for the students learning process AND their correctness checking process. It is brilliant! It is like teaching someone to swim, ride a bike or in this case know that what you know is correct.
I help adults solve problems in their code. I'm going to try this approach and see if it works in that context.
Hopefully it will. Although I guess in some cases the issue is reversed parameters to a function, or whatever, where the underlying knowledge is weak (or faulty).
Whenever I’m making a plan, or figuring something out, be it engineering or code or physics or whatever, I rubber duck - that is, I explain what I’m doing and why, verbally, in exhaustive detail, to nothing and nobody in particular (for I do not have a rubber duck), and ask questions of myself that a rubber duck would ask. I often catch a snag where I would otherwise have thought my scheme flawless.
Simple editors that don't point out any problems with code formatting or syntax, and compilers with poor diagnostics like "error: this file has a syntax or type error somewhere" ... turn out to be good parents!
Helping people learn something is a lot like debugging software. Usually the problem is that there's an assumption that is wrong. You need to ask questions and check the answers to find out which assumption is wrong. You can even use bisection between input and output to find the wrong assumption in logarithmic
time :), but if there aren't that many steps it's less confusing to go step-by-step from input to output.
Once you find it - fixing it is easy. Provide simple counterexamples showing why it's wrong, and watch the person solve the problem on their own.
I like this advice, but my daughter has been taught to do basic math "intuitively".
E.g. for multidigit addition instead of adding up units then carry, they are taught to just add everything up. Learning the carry method only happens later.
She can't explain how she does it, and I can't either, it's been a frustrating experience.
I hope I can still be useful when she gets to harder stuff that requires more steps :)
Is this a mistake in a proof or a mistake in a calculation?
Let's say that the distinction here is how we establish that an answer is correct. A proof is correct if it typechecks (no really). A computation is correct if it faithfully executes the algorithm.
This distinction is more than a little artificial, but in practice it matters a lot.
Yes, it’s very disheartening to see math teachers not understand how to check your work.
There are algorithms (and heuristics) that have been known for centuries to verify (or estimate) check most results from arithmetic to calculus.
It’s as if programmers were not taught how to do logging or assertions.
In math there are two types of errors, not understanding the problem and mistakes. Redoing the work won’t help at all for misunderstanding or lack of understanding and has only a marginal chance of catching the mistake.
Try very hard not to be patronising. Kids struggling with maths can be super sensitive. They can misinterpret a smile as a smirk. If they had the courage to seek help, don't belittle them. For example a phrase as simple as "that's easy" can be a huge burden.
Beware misunderstood questions. If you seek to help by asking them questions be mindful if they do not understand the intent of your question, you may add to their burden not reduce it.
Some set work requires a specific method be shown to be used. It's fine to show them other methods, but you may have to help them project the answer back into the one they will be assessed in.
If they only want debugging, then help them find the bug, the mistranscribed value, the divide by zero, the specific mistake. If they have no insight into the problem then more exploratory help may be useful.
Time out can help. Stress hormones don't make maths any easier.